� Lecture 24 8.251 Spring 2007 Lecture 24 - Topics • Dp-brane • Parallel Dp’s X˙− ± X−� = 21 α� 2p1 + (X˙I ± XI �)2 (1) X˙I ± XI � = √2α� � αI exp(−in(τ ± σ)) (2)nn∈Z 1 � 1 ∞2p + p− = α� 2 α0I αI 0 + α−nαI + a) (3)n n=1 Dp-brane x0, x1, . . . , xp: Coordinates on brane x+, x−, xi, i = 1, 2, 3, . . . , p ⇒ (p + 1) − 2 values. N N coordinates. xp+1, . . . , xd: Normal to brane ax , a = p + 1, . . . , d ⇒ (d − p) values. DD coordinates. 1� ��� � � Lecture 24 8.251 Spring 2007 Thus: Equation (1) becomes: 1 1 2α� 2p+ [( ˙x i ± x i�)2 + ( ˙x a ± x a�)2] Equation (2) holds. Equation (3): 1 � 1 ∞� 2p + p− = α� 2 α0I α0 I + α−nαnI + a n=1 Xa(τ, σ): Xa(τ, 0) = Xa(τ, π) = X a , a scalar αa n Xa(τ, σ) = X a + √2α� � 1 ne−inσ sin nσ n=0 Xa� ± ˙= √2α� � Xa αa e−in(τ ±σ) nn=0 [αa , αb ] = mδm+n,0δab m n2p + p− = 1(α�p i p i + ∞(α−inα+in + α−anαa ) − 1)α� n=1 n1 M2 = (−1 + N⊥)α� N⊥ = �Nlongitudinal + Ntransverse Nl = ∞ αi αi �n=1 −n n Nt = ∞ αa αa n=1 −n n Ground state: M2 = − α1 � , State �p+, pi > tachyons, gives rise to ψ(τ, p+, pi). ap = 0. 2��� ��� � �� �Lecture 24 8.251 Spring 2007 Where do these fields live? Reasonable to say, on the brane. Right number of acoordinates p − 1 dim. space. But where did the x go? Next state: αi p+, pi , M2 = 0. −1 (p + 1) − 2 of these states. What are they? Photon states! iAlso, αa lives on brane but has xa index. Nothing to do with spacetime. +p , p−1 Inert scalar states. d − p massless scalars. Physical interpretation: Represent possible excitations casting 0 energy and 0 momentum, displacing the brane. (See QFT theory of Goldstone). Parallel Dp’s Know everything about strings A and B. New problem: strings C and D. (note C =� D since orientation matters) A = [1, 1], B = [2, 2], C = [1, 2], D = [2, 1]. In general, [i, j] with σ = 0 ∈ Di, σ = π ∈ Dj . (Before, always talking about same D-branes, so there was an implicity [1, 1] always.) 11 a a 2 − X1 )σ + √2α� a αa ne−inσ sin nσXa(τ, σ) = X (X+i π n n=0 = √2α�Xa� αa nn=0 3 e−inτ cos(nσ)��� ��� � � Lecture 24 8.251 Spring 2007 √2α�αa 0 = π 1(X¯ 2 a − X¯ 1 a) Alternatively, write as: α0 a 1 a a √2α� =2πα0�(X2 − X1 ) 1 12p + p− = α� (αi p i p i +2 α0 aαa 0 + N⊥ − 1) 1 1 αaM2 = α� (N⊥ − 1) + 2α� 0 α0 a 1 (N⊥ − 1): Contribution of Quantum Oscillation α� 21 α� α0 aα0 a = (T0(X2 a − X1 a))2: Contribution of Tension×Length Now M2 quantized for any sector, but can choose sectors. i[1, 2] , pi[1, 1]Ground states: (we know how to handle this one) + +p , p p, ��p��State αi −1 State αa −1 +, pi[1, 2] D-branes separate, so now not massless photons⇒+, pi[1, 2] same mass. Always a scalar state. pSuppose have 3 branes: 4Lecture 24 8.251 Spring 2007 String A = [i, j] String B = [j, k] Can interact to form one string between Di and Dk: [i, j] × [j, k] = [i, k]. (Then Dj doesn’t notice the string anymore). Theory of interacting gauge fields. Dp brane parallel to Dq Coordinates have split into common D, common N and split N D.